Question

Найдите значение выражения

Answer 1

$sin2alphacos5alpha-sinalphacos6alpha=2sinalphacosalphacos5alpha-sinalphacos6alpha=\ \ =sin alpha(2cos alphacos5alpha-cos6alpha)=sinalpha(cos 4alpha+cos6alpha-cos6alpha)=\ \ =sinalphacos4alpha=sinalpha(2cos^22alpha-1)=sinalphaBig(2cdot (1-2sin^2alpha)^2-1Big)=\ \ =sinalpha(2-8sin^2alpha+8sin^4alpha-1)=aBig(8a^4-8a^2+1Big)$

$cos7alphacos4alpha-cos8alphacos3alpha=dfrac{1}{2}Big(cos3alpha+cos11alphaBig)-dfrac{1}{2}Big(cos 5alpha+\ \ +cos11alphaBig)=dfrac{1}{2}Big(cos3alpha-cos5alphaBig)=dfrac{1}{2}cdot 2sindfrac{3alpha+5alpha}{2}sindfrac{5alpha-3alpha}{2}\ \ =sin4alphasinalpha=2sin2alphacos2alphasinalpha=4sin^2alphacosalpha(2cos^2alpha-1)\ \ =4(1-cos^2alpha)cosalpha(2cos^2alpha-1)=4a(1-a^2)(2a^2-1)$